This can be a tricky city for visitors since the downtown area is fairly small and hotels in the center can get away with charging a fortune. There are some cheap hotels in non-touristy parts of town, but honestly you are much better off in Virginia in the suburbs a bit, and public transportation does a great job of quickly getting you into the center during the day. On the other hand, cheap flights to Washington DC are usually easy to find due to 3 major airports in the region.
Hotels in Washington DC tend to be busy all year long, with prices that match New York City most of the time. This is a big tourist town and also huge for business travelers, and the cheapest time of year for hotels is January through March. High season is spring and autumn, and summer usually brings prices down a bit. Good deals can be had on weekends, but usually only at the business-oriented places.